In Mathematics, It Often Takes a Good Map to Find Answers
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In Mathematics, It Often Takes a Good Map to Find Answers
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Re: In Mathematics, It Often Takes a Good Map to Find Answers
#2> But imagine how poetic it would have been if the technology for constructing such a machine had been available to da Vinci all along.
Very poetic indeed.
Most of entrepreneurship is applying known models to new areas. Intellectually not nearly as stimulating or hard as theoretical math, but the shape and form looks similar - you do not know if a solution exists, you do not know if a problem really exists.
What's funny to me is that, since it's usually applications of engineering, the technology is almost always there. It's a matter of tinkering a collection of things the right way.
I ditched a career in Physics to start a company long back. This post made me think I probably haven't lost much :-)
Re: In Mathematics, It Often Takes a Good Map to Find Answers
#3I highly recommend Bill Thurston’s gem of an article On proof and progress in mathematics
https://arxiv.org/abs/math/9404236
Talks about the human aspect of pursuing mathematical research, how they shape the attitude of the field towards a problem abs are crucial in progressing towards knowledge. Should be very readable for everyone; no formal math as such.
Re: In Mathematics, It Often Takes a Good Map to Find Answers
#4The article is sparse on what a detailed map for mathematics would look like and merely points out that some topics have related techniques for solving them.
I don't think a map for math techniques is feasible, but a map relating topics via mathematical steps is possible in Physics [1]. (Disclaimer: I'm the author of that map for Physics.) I think the reason that a map in Physics is feasible is Physicists do not use math techniques in the way mathematicians do, and the objectives are different.
Re: In Mathematics, It Often Takes a Good Map to Find Answers
#5The article is sparse on what a detailed map for mathematics would look like and merely points out that some topics have related techniques for solving them. I don't think a map for math techniques is feasible, but a map relating topics via mathematical steps is possible in Physics [1]. (Disclaimer: I'm the author of that map for Physics.) I think the reason that a map in Physics is feasible is Physicists do not use…
It seems to me that your map is far too detailed to use practically. You spell out every algebraic step, including stuff as simple as "divide both sides by T", so that deriving f = 1/T from T = 1/f takes about 10 nodes. This is like building a model train to a larger scale than an actual train -- what is the use?
Education research tells us that what you actually want to do is the exact opposite: chunk as much as possible. You should learn algebra separately, and then use your preexisting knowledge of algebra to group f = 1/T and T = 1/f into one conceptual node. If you need 10 nodes every time something that basic is done, then your map will contain a vast amount of redundancy and be too large to use to get anywhere...
Re: In Mathematics, It Often Takes a Good Map to Find Answers
#6The article is sparse on what a detailed map for mathematics would look like and merely points out that some topics have related techniques for solving them. I don't think a map for math techniques is feasible, but a map relating topics via mathematical steps is possible in Physics [1]. (Disclaimer: I'm the author of that map for Physics.) I think the reason that a map in Physics is feasible is Physicists do not use…
It seems to me that your map is far too detailed to use practically. You spell out every algebraic step, including stuff as simple as "divide both sides by T", so that deriving f = 1/T from T = 1/f takes about 10 nodes. This is like building a model train to a larger scale than an actual train -- what is the use? Education research tells us that what you actually want to do is the exact opposite: chunk as much as pos…
I agree that navigating a map of Physics at the very lowest level would not enlighten any student or researcher. My expectation in mapping atomic steps for a wide swath of the domain might enable insights not otherwise accessible.
The chunking of atomic steps is what enables leaps in understanding. The mapping process starts with understanding each step.
Re: In Mathematics, It Often Takes a Good Map to Find Answers
#7The difficulty in coming up with a good map of mathematics is summarized by this quote by Banach:
"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."
Re: In Mathematics, It Often Takes a Good Map to Find Answers
#8Earlier quoted context omitted.
It seems to me that your map is far too detailed to use practically. You spell out every algebraic step, including stuff as simple as "divide both sides by T", so that deriving f = 1/T from T = 1/f takes about 10 nodes. This is like building a model train to a larger scale than an actual train -- what is the use? Education research tells us that what you actually want to do is the exact opposite: chunk as much as pos…
I agree that navigating a map of Physics at the very lowest level would not enlighten any student or researcher. My expectation in mapping atomic steps for a wide swath of the domain might enable insights not otherwise accessible. The chunking of atomic steps is what enables leaps in understanding. The mapping process starts with understanding each step.
Well, I recommend doing a concrete, nontrivial derivation from start to finish just to see how this approach scales. As a basic example that is typically covered in about half a page in books, try doing a full derivation of the wave equation for a wave on a string. I would bet that once you set up the 1000 nodes required to do this, you'll be completely exhausted, and moreover will have gotten no new insight! If you're not tired yet, try deriving the equation describing waves on a stiff rod -- it'll take at least 1500 nodes, most of which will be exactly the same as the ones for the wave equation.
Furthermore, this excessive mathematical structure hides the physical assumptions that really drive the validity of these equations. A real string doesn't actually obey the wave equation perfectly. The reason has to do with physical aspects of the string itself, not minutiae in the mathematical derivation of the wave equation. I can't think of an example where progress in physics was stalled because somebody tried to divide both sides of an equation by T and failed...
Re: In Mathematics, It Often Takes a Good Map to Find Answers
#9I liked this article. One notable point that felt like it was missing in the article is that the Prime Number Theorem, that the count of primes grow like (n / ln n) was provided such a map by Riemann in the letter in which he put forward his infamous eponymous hypothesis. That letter introduced the idea of using analysis to the Prime Number Theorem, extending the groundbreaking work of Riemann's friend Dirichlet who introduced the world to analytic number theory in Dirichlet's Theorem on the infinitude of primes in arithmetic progressions. It would take nearly half a century for mathematicians to digest the application of Fourier Analysis put forward by Riemann, and the proof of the Prime Number Theorem came only in the early 1900's. By then the analytic machinery would have been more commonly taught -- probably largely due to the advent of electrical engineering.
Erdos and Selberg eventually put out fully arithmetic proofs of the Prime Number Theorem. And generally the helicopter analogy from the article probably doesn't apply so well to mathematics because you can probably always reduce theories and encapsulate all the dependent proofs to arithmetic first principles, but of course you already have the map.
Recently the proofs of the Sensitivity Conjecture by Hao Huang and of the Bounded Gaps Between Primes by Yitang Zhang surprised mathematicians in how little new machinery these seemingly intractable problems required -- in the case of Zhang application of "hard work" on top of GPY and Hao Huang, a single clever insight.
Re: In Mathematics, It Often Takes a Good Map to Find Answers
#10The article is sparse on what a detailed map for mathematics would look like and merely points out that some topics have related techniques for solving them. I don't think a map for math techniques is feasible, but a map relating topics via mathematical steps is possible in Physics [1]. (Disclaimer: I'm the author of that map for Physics.) I think the reason that a map in Physics is feasible is Physicists do not use…
It seems to me that your map is far too detailed to use practically. You spell out every algebraic step, including stuff as simple as "divide both sides by T", so that deriving f = 1/T from T = 1/f takes about 10 nodes. This is like building a model train to a larger scale than an actual train -- what is the use? Education research tells us that what you actually want to do is the exact opposite: chunk as much as pos…
Here's 2+2=4: https://twitter.com/dd4ta/status/1050433711416721408
I recently asked a related question regarding proof maps and quantifying their similarity/distance, but didn't get any answers: https://math.stackexchange.com/questions/3482135/are-there-p...